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Jul 22 STRANGE LOOPS

The seventeenth chapter of Abstruse Geometry concerns the mathematical problem of snakes swallowing their own tails. Following in the footsteps of a lost treatise by Athanasius Kircher, the tome describes a mysterious knot known as an “ourobohedron,” which is composed entirely of snakes engaged in varying degrees of autophagy. What it lacks in mathematical rigor, it makes up for in curiosity. The figure is described as follows:

1. The bottommost vertex of an ourobohedron consists of a serpent swallowing a serpent swallowing its own tail before swallowing its own tail.

2. The lateral vertices of an ourobohedron consist of six individual serpents.

The first swallows the outermost serpent of the bottommost vertex headfirst before swallowing its own tail.

The second swallows the outermost serpent of the bottommost vertex tailfirst before swallowing its own tail.

The third swallows the first headfirst before swallowing its own tail.

The fourth swallows the first tailfirst before swallowing its own tail.

The fifth swallows the second headfirst before swallowing its own tail.

The sixth swallows the second tailfirst before swallowing its own tail.

3. Similar to its opposite pole, the topmost vertex of an ourobohedron consists of a serpent swallowing a serpent swallowing its own tail before swallowing its own tail. However, the outermost serpent of the topmost vertex is the innermost serpent of the bottommost vertex, and the outermost serpent of the bottommost vertex is the innermost serpent of the topmost vertex.

4. An ourobohedron consists of exactly eight serpents. Any serpent which attempts to swallow an ourobohedron is, in turn, swallowed whole by the ourobohedron, and as such, cannot be integrated into the knot.

5. Ouroborization is a unique property of serpents. If eels, lampreys, or other legless creatures attempt to swallow their own tails, they simply cease to exist. As such, no other creature can be integrated into an ourobohedron without compromising the integrity of the entire knot.

6. When an ourobohedron sheds its own skin, that which is discarded is also a complete ourobohedron.

Much to the chagrin of scholars, while the ourobohedron is described thoroughly within, it is not actually pictured. The author gives a suspect excuse for this omission: he claims that whenever he attempted to draw it, he could not prevent the incomplete figure from swallowing his pen whole.

The problem of the individual ouroboros has been discussed in great detail.